Math 5334: Differential Geometry

 

Course Description: This course covers both the differential and the geometric parts of differential geometry.  The differential part includes manifolds, the tangent bundle of a manifold, differential forms, tensors, and Riemannian metrics.  All of this is used in formulating the geometric part, which includes curvature and torsion of space curves, Gaussian curvature of surfaces, the Gauss-Bonnet Theorem, covariant derivatives, parallel transport, and the Riemann curvature tensor on manifolds of higher dimension.

 

List of Topics:

1. Manifolds

    Examples of manifolds

    Coordinate charts

    Transition functions between coordinate charts

    Atlases on manifolds 

    Differentiability of functions between manifolds

    The Jacobian

    The chain rule  

 

2. The Tangent Bundle

    Description of vectors as linear operators

    Transformations of vectors under changes of coordinates

    Vector fields

 

3. The Cotangent Bundle

    Differential 1-forms

    Transformations of differential forms under a change of coordinates

   

4. Tensors

    Covariant and contravariant tensors

    Raising and lowering of indices (via a Riemannian metric)

   

5. Inner Products

    The matrix of an inner product with respect to a basis

    Riemannian metrics on manifolds

 

6. Curves in the Plane and in Space

    Curvature and torsion of plane curves and space curves

    The Serret-Frenet formulas

   

7. Curvature of Surfaces in Space

    Gaussian curvature

    The first and second fundamental forms

    The Theorema Egregium

   

8. Differential Geometry on Higher-dimensional Manifolds

    Riemann curvature tensor (its definition and meaning)

    Torsion tensor (its definition and meaning)

    Covariant derivatives and the Levi-Civita connection

    Parallel transport

 

9. The Gauss-Bonnet Theorem

 

References:

1. Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volumes I and II (and Vol. III for the Gauss-Bonnet Theorem)

2. Manfredo Do Carmo, Differential Geometry of Curves and Surfaces

3. Michael Spivak, Calculus on Manifolds

4. M. Nakahara, Geometry and Physics